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Metamath Proof Explorer

Theorem hbequid

Description: Bound-variable hypothesis builder for x = x . This theorem tells us that any variable, including x , is effectively not free in x = x , even though x is technically free according to the traditional definition of free variable. (The proof does not use ax-c10 .) (Contributed by NM, 13-Jan-2011) (Proof shortened by Wolf Lammen, 23-Mar-2014) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	hbequid	⊢ ( 𝑥 = 𝑥 → ∀ 𝑦 𝑥 = 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		ax-c9	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑥 → ( ¬ ∀ 𝑦 𝑦 = 𝑥 → ( 𝑥 = 𝑥 → ∀ 𝑦 𝑥 = 𝑥 ) ) )
2		ax7	⊢ ( 𝑦 = 𝑥 → ( 𝑦 = 𝑥 → 𝑥 = 𝑥 ) )
3	2	pm2.43i	⊢ ( 𝑦 = 𝑥 → 𝑥 = 𝑥 )
4	3	alimi	⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ∀ 𝑦 𝑥 = 𝑥 )
5	4	a1d	⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( 𝑥 = 𝑥 → ∀ 𝑦 𝑥 = 𝑥 ) )
6	1 5 5	pm2.61ii	⊢ ( 𝑥 = 𝑥 → ∀ 𝑦 𝑥 = 𝑥 )